Multiple choice

Two pillars of equal height stand at a distance of $100$ metres. $A$t a point between them the elevation of their tops are found to be $30^{ }$ and $60^{0}$ Then height of the each pillar in metres is

  1. $25\sqrt{3}$
  2. $20\sqrt{3}$
  3. $50\sqrt{3}$
  4. $35\sqrt{3}$
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A Correct answer
Explanation

Let height be h. Distances are h/tan(30) and h/tan(60). h*sqrt(3) + h/sqrt(3) = 100. h(3+1)/sqrt(3) = 100. 4h = 100*sqrt(3). h = 25*sqrt(3).

AI explanation

Let the point be x meters from the first pillar, making its distance from the second pillar 100 minus x meters. Using the tangent ratio, tan(60) = h / x and tan(30) = h / (100 - x). Since tan(60) is sqrt(3) and tan(30) is 1 / sqrt(3), the equations become h = x * sqrt(3) and h = (100 - x) / sqrt(3). Equating the two expressions for h and multiplying by sqrt(3) gives 3x = 100 - x, so x = 25. Substituting x back into the first equation gives h = 25 * sqrt(3). The result is 25 * sqrt(3).

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